Analysis of singularity of solutions to the Navier-Stokes equations

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We prove regularity of weak solutions to the Navier-Stokes equations under the conditions on the pressure. In particular, we exelude the possibility of blow-up in the case that the pressure is small enough in scaling invariant spaces based on Lorentz spaces. This result can be applied to MHD equations if the magnetic field in addition to the pressure belongs to the scaling invariant spaces based on the Lorentz spaces with small norms.